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Theorem frgrncvvdeqlem1 28659
Description: Lemma 1 for frgrncvvdeq 28669. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
Hypotheses
Ref Expression
frgrncvvdeq.v1 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlem1 (𝜑𝑋𝑁)

Proof of Theorem frgrncvvdeqlem1
StepHypRef Expression
1 frgrncvvdeq.xy . . . 4 (𝜑𝑌𝐷)
2 df-nel 3052 . . . . 5 (𝑌𝐷 ↔ ¬ 𝑌𝐷)
3 frgrncvvdeq.nx . . . . . 6 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2832 . . . . 5 (𝑌𝐷𝑌 ∈ (𝐺 NeighbVtx 𝑋))
52, 4xchbinx 334 . . . 4 (𝑌𝐷 ↔ ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
61, 5sylib 217 . . 3 (𝜑 → ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
7 nbgrsym 27728 . . 3 (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
86, 7sylnibr 329 . 2 (𝜑 → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
9 frgrncvvdeq.ny . . . 4 𝑁 = (𝐺 NeighbVtx 𝑌)
10 neleq2 3057 . . . 4 (𝑁 = (𝐺 NeighbVtx 𝑌) → (𝑋𝑁𝑋 ∉ (𝐺 NeighbVtx 𝑌)))
119, 10ax-mp 5 . . 3 (𝑋𝑁𝑋 ∉ (𝐺 NeighbVtx 𝑌))
12 df-nel 3052 . . 3 (𝑋 ∉ (𝐺 NeighbVtx 𝑌) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
1311, 12bitri 274 . 2 (𝑋𝑁 ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
148, 13sylibr 233 1 (𝜑𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1542  wcel 2110  wne 2945  wnel 3051  {cpr 4569  cmpt 5162  cfv 6432  crio 7227  (class class class)co 7271  Vtxcvtx 27364  Edgcedg 27415   NeighbVtx cnbgr 27697   FriendGraph cfrgr 28618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-nbgr 27698
This theorem is referenced by:  frgrncvvdeqlem7  28665  frgrncvvdeqlem8  28666  frgrncvvdeqlem9  28667
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